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http://functions.wolfram.com/12.06.22.0006.01
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FourierTransform[Abs[t]^\[Alpha] Sign[t], t, x] ==
I Sqrt[2/Pi] Abs[x]^(-1 - \[Alpha]) Cos[(Pi \[Alpha])/2]
Gamma[1 + \[Alpha]] Sign[x] /; Re[\[Alpha]] > -1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", "t", "]"]], "\[Alpha]"], RowBox[List["Sign", "[", "t", "]"]]]], ",", "t", ",", "x"]], "]"]], "\[Equal]", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", "x", "]"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]], " ", RowBox[List["Sign", "[", "x", "]"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], " ", ">", RowBox[List["-", "1"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msub> <mi> ℱ </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> t </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mi> α </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo>  </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> <mo> ⁢ </mo> <msup> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> x </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mrow> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> α </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> FourierTransform </ci> <apply> <times /> <apply> <power /> <apply> <abs /> <ci> t </ci> </apply> <ci> α </ci> </apply> <apply> <ci> Sign </ci> <ci> t </ci> </apply> </apply> <ci> t </ci> <ci> x </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <abs /> <ci> x </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> α </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Sign </ci> <ci> x </ci> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> α </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Abs", "[", "t_", "]"]], "\[Alpha]_"], " ", RowBox[List["Sign", "[", "t_", "]"]]]], ",", "t_", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox[RowBox[List["Abs", "[", "x", "]"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Alpha]"]], "]"]], " ", RowBox[List["Sign", "[", "x", "]"]]]], "/;", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
